■マルコフ方程式の話(その11)

 arccosh(x)=log(x+√(x^2−1))

=log2x−Σ(2n−1)!!/2n(2n)!!・1/x^2n

の方が使いやすいと思われる.

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 arccosh(3x/2)=log(3x)−Σ(2n−1)!!/2n(2n)!!・1/(3x/2)^2n

 arccosh(3y/2)=log(3y)−Σ(2n−1)!!/2n(2n)!!・1/(3y/2)^2n

 arccosh(3z/2)=log(3z)−Σ(2n−1)!!/2n(2n)!!・1/(3z/2)^2n

f(x)+f(y)=log(9xy)−Σ(2n−1)!!/2n(2n)!!・1/(3x/2)^2n−Σ(2n−1)!!/2n(2n)!!・1/(3x/2)^2n

=log(9xy)−Σ(2n−1)!!/2n(2n)!!{1/(3x/2)^2n+1/(3y/2)^2n}

=log(9xy)−Σ(2n−1)!!/2n(2n)!!{(3x/2)^2n+(3y/2)^2n}/(9xy/4)^2n

f(z)=log(3z)−Σ(2n−1)!!/2n(2n)!!・1/(3z/2)^2n

  x^2+y^2+z1^2=3xyz1

  x^2+y^2+z2^2=3xyz2

 z1=xy/2・{3+(9−4/x^2−4/y^2)^1/2}

 z2=xy/2・{3−(9−4/x^2−4/y^2)^1/2}

 z=z1+z2=3xy

 z1z2=(xy/2)^2(4/x^2+4/y^2)=x^2+y^2

f(z1+z2)=log(3(z1+z2))−Σ(2n−1)!!/2n(2n)!!・1/(3(z1+z2)/2)^2n

=log(9xy)−Σ(2n−1)!!/2n(2n)!!・1/(9xy/2)^2n

1/(9xy/2)^2n={(3x/2)^2n+(3y/2)^2n}/(9xy/4)^2n

1/2^2n={(3x/2)^2n+(3y/2)^2n}

1={(3x)^2n+(3y)^2n}であればよい???

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